RAIROANALYSE NUMÉRIQUE

JUHANI PITKÄRANTAOn a mixed finite element method for theStokes problem in R3

RAIRO – Analyse numérique, tome 16, no 3 (1982), p. 275-291.<http://www.numdam.org/item?id=M2AN_1982__16_3_275_0>

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R A I R O Analyse numérique/Numencal Analysis(vol 16, n°3, 1982, p 275 a 291)

ON A MIXED FINITE ELEMENT METHODFOR THE STOKES PROBLEM IN U3 (*)

by Juhani PITKARANTA (*)

Communicated by P G CIARLET

Abstract — We prove an error estimatefor a mixed flnite element methodfor solving the Stokesproblem on a rectangular domain in U3 The scheme is based on piecewise trihnear velocities andpiecewise constant pressure on a uniform rectangular gnd

Résume — On établit une estimation de Verreur pour une methode d''éléments finis mixtes pourle problème de Stokes sur un domaine rectangulaire de M3 Le schema met en oeuvre des vitessestnhneaires par morceaux et une pression constante par morceaux sur un maillage rectangulaireuniforme

1. INTRODUCTION

One of the simplest ways of discretizing the Stokes équations on a rectan-gular domain in Rn is to apply the finite element technique with continuous,piecewise multilinear velocities and piecewise constant pressure on a rectan-gular grid. The resulting fînite différence équations resemble those of theclassical Marker — and — Cell method [4]. In two dimensions the methodhas been used successfully also on irregular meshes, cf. [10].

From a theoretical point of view, the above fînite element scheme fallsinto the category of mixed methods, which can be analyzed along the linesof Babuska [1] and Brezzi [2]. The analysis was recently carried out in thetwo-dimensional case [7]. It was shown that although the method is not uni-formly stable in the classical sensé of [1, 2], a weaker stability estimate holdswhich yields optimal convergence rates for the velocities in H 1(Q) and L2(Q),provided that the exact solution is sufficiently regular.

(*) Received in April 1981l1) Institute of Mathematics, Helsinki University of Technology, SF-0150 Espoo 15, Finlande.

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276 J PITKARANTA

In this paper we analyze the three-dimensional scheme where the velocitiesare approximated by piecewise tnlmear functions The analysis proceedsfollowing closely the lmes of [7] In particular, we establish a weak Babuska-Brezzi-type stabihty estimate for the pressures and combine this with certainsuperapproximation properties for the velocities As in two dimensions, weare able to prove that the velocities converge with the optimal rate O(h) mH1(Q), if the exact solution is sufficiently smooth We also state the three-dimensional analogues of the L2-estimates proved in [7] for the velocities andfor the pressures smoothed m an appropnate way

Due to the fact that the stabihty estimate we can prove is weaker than mtwo dimensions, we end up requirmg relatively high regulanty on the exactsolution, in order to be able to balance the weak stabihty with superapproxi-mation results Only the case of a regular mesh is considered, a constraintthat seems to play an essential role in the analysis

The plan of the paper is as follows In section 2 we state the problem anddefine lts finite element discretization Section 3 is devoted to the error ana-lysis

Throughout the paper we dénote by Wm P(Q), Q c [R3, m ^ 0, 1 ^ p < oo,the usual Sobolev spaces with the norms

\ÜP

\\v\\mp = ( £ \v\!p\

where \.\lp dénote the semmorms

[=i j Q

dx\ dxJ2 dx\

dxx dx2

Here we omit to indicate the domain with a submdex, since it will be thesame throughout the paper For non-integral s ^ 0, Wsp(Q) is defined asusual by interpolation For p = 2 we set Hm(Q) = Wm 2(Q), | . |w = | • L 2and || . ||w = || . L 2 A s usual> Hl

0{ü) dénotes the completion of C0°°(Q) mthe norm | | . || x

The same notation will be used for the corresponding (semi) norms in[Wm>p(Q)]3 The scalar products in L2(Q) or [L2(Q)]3 will be denoted by

Finally, by C or C3 we dénote positive constants, possibly different at dif-ferent occurrences, which may depend on the domam Q considered but noton any other parameter to be mtroduced unless mdicated explicitly We alsodénote by Pk the set of polynomials in three variables of degree at most k

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THE STOKES PROBLEM IN R3 277

2. THE PROBLEM AND ITS DISCRETIZATION

Let Q be a rectangular domain in IR3 : Q = {x = (x1,x2,x3) eU3,xt e (0, at\ i = 1, 2, 3 }. We consider the Stokes problem for an incompressiblefluid with viscosity equal to one :

- Au + Vk = f in Q ,

divu = 0 in Q, (2.1)

u = 0 on öQ,

1Idx = 0.

Here w = {u1,u2, u3) is the velocity of the fluid and X is the pressure, whichwe normalize to have the zero mean value. For simplicity we consider onlythe homogeneous Dirichlet boundary condition.

Let C ° be a uniform partitioning of Q into rectangular subdomains of sizeh1 x h2 x /z3, i.e.,

e,? = { Kljk : i = 1,..., ml57 = 1,..., m2, k = 1,..., m3 } ,

£;jfc = { (*i, ^2, ^3) e IR3 : (f - 1) ^ < xx < * ! ,

(7 - 1) fc2 < *2 < 7*2» (fc - 1) /i3 < ^3 < ^ 3 } >

where m£ = aj^ are integers. We assume that hu h2 and /i3 depend on themesh parameter h in such a way that hjh is bounded from below and fromabove by constants independent of h.

Let Ch be a partitioning of Q obtained by dividing each Kljk e Q° intoeight equal 3-rectangles :

Ch = {Aijk : i = l,...92ml9j = 1 2 m 2 , fc = 1,..., 2 m3 } ,

AiJk = { x G R3 ; (Ï - 1) hx/2 < xx < ihjl,

(j - 1 ) h2/2 <x2< j h 2 / 2 9 (k - 1 ) h3/2 <x3< kh3/2 } .

We associate to Ch the following finite element spaces :

Sh = {ve Hh(O) : v \Axjk is trilinear VAijk e Ch }

Qh = { \i e L 2 ( Q ) : ^ \Aijk i s c o n s t a n t V A 0 * e C h } .

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278 J PITKARANTA

Setting Vh = (Sh)3 we can now defme a finite element method for the solu-

tion of (2 1) as Fmd (uft, Xh) G Vh x Qh such that

(Vnfc, VÜ) - (Xh9 div t>) = (ƒ, v) VveVh) (2 2a)

(divufc,ji) = 0 n i é e * f (2 2b)

This set of équation does not have a unique solution (see section 3 below)To make the solution unique, it is customary to replace (2 2b) by

e(A.h,li) + (divt<fc,n) = O V^iGÔ,,, (2 26')

where 8 > 0 is a small parameter The perturbed System (2 2a)-(2 2b') nowhas a unique solution, as is easily seen by setting v = uh \x = Xh Upon eli-minating Xh from the perturbed System one obtams for uh the équation

(Vu,, Vi>) + \ (div uh, div t;)* = ( ƒ ü) \fv G F , (2 3)o

where (., .)* indicates that the inner product is evaluated by first taking theaverage of div uh and div v over each AIjfc G Ch Eg (2 3) may also be regardedas a penalty method where the so-called sélective reduced intégration (cf [8])is apphed

In the analysis below we will only treat the unperturbed scheme (2 2a, b)It is possible to show (see [7] for details) that the results also hold for thescheme (2 2a, b ), provided that 8 ^ Ch2

3 ERROR ANALYSIS

We will first introducé a special orthogonal basis for the space Qh Thebasis consists of the functions ^ljkh i = 1, , ml9 j = 1, , m2, k — 1, , m3,/ = 1, ,8 defined as follows The support of each £IjklJ / = 1, , 8, is con-tained in Kljk e Ch°, and on each subrectangle AVlV2V3 c £ u k , AV1V2V3 G Ch,the functions ^ljkh / = 1, ,8 , attain the value ± 1 according to the followingrule

if x 6 AV1V2V3 e Ch5 AVlV2V3 c Kljk e Q°

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THE STOKES PROBLEM IN R3 279

Any [i e Qh has the unique représentation

H = Z a y « £>ijkl •hJ,k,l

Here and below we sum i,j, k and / from I t o m 1 , m 2 , ra3 and 8, respectively,unless noted otherwise.

We introducé the following subspaces of Qh :

Nh = { \i e Qh : (m div i;) = 0 VveVh}

One can verify by simple computation that Nh consists of the linear combina-tions of functions \|/, cpp i = 1,..., ml5 6;, 7 = 1,..., m2 and pfc, fc = 1,..., m3,defined as follows :

q>,(x)= f ( - i y + \ x6A I J k6C f c

t 0 , otherwise

0,(x)= f ( - l ) I + k , xeA l j f ceC f c

[ 0 otherwise

Pk(x) = j ( " 1)I + ' , X6A l j k6C f c

[ 0 , otherwise .

Taking into account the relation ]T q>, = Y, 6j = X! Pk' w e conclude easily

that dim (Nh) = 2(m1 + m2 + m3) - 1.The space Njj- can now be characterized as

Nh = Zijk

E auk5 = Z aljk8 = 0 , ï = 1,..., mx ,J,k j,k

Z aufc6 = Z aufc8 = 0 , 7 = 1,..., m2 ,l,fc I.fc

Z *uki = Z aufc8 = 0 , k = 1,..., m3 l .

Remark : The solution of (2.2) is not unique, since if (uh, Xh) is a solution,then so is (uh, Xh + \i) for any |i e Nh. However, if we require that Xh e N^~

vol 16, n° 3, 1982

280 J. PITKÂRANTA

then the solution is unique. Note also that if (wh, A,h) is the solution of theperturbed problem (2. 2a, b% then Xh e N^. D

We will supply Qh with a special mesh-dependent semi-norm, the meaningof which will be clarified by Lemma 3.1 below. We define

1=1 1=5

r — ZJ aijk/ SijkZ >i,J,kJ

where

and mz 1

J = l

mi~ 1

n i - 1

W l l 1

Y (h a

5"§('ffl2- 1

m

+ t

+ z

•yk5

„6

Ik

1 - 1

Zm2

vi,J+l,k5

- a I + 1 , j k 7 ) 2

n i 3 - 1

- 1 1113- 1

L Z (ai^

m3~ 1

fc=l i,J

m 3 ~ 1! + I S(

k = l i,j

mi~ 1

i.j*8 - a . j+1

k8 ~" a i+l ,jk8

c8 ~ ai ,j+l,k8

a»jk5

,w + «,+

We now prove a stability estimate of Babuska-Brezzi (cf. [1, 2]) type.

LEMMA 3 . 1 : There are the constants Cx and C2 such that

for all \ieQh with (ju, 1) = 0.

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THE STOKES PROBLEM IN R3 281

In the proof we need the following analogue of Lemma 3.1, obtained byreducing the space Qh to consist only of functions that are constant on each

LEMMA 3.2 : Let [xx = £ ocljkl £,ljkl, with (\il9 1) = 0. Then there is a cons-ijk

tant C such that

Proof : Given \ix as in the lemma, there exists (cf. [5]) z G [if J (^)] 3 suchthat div z = |ix in Q and

II* 111 <c\\ M o .

We then define zh e Vh by requiring

zh(P) = wh(P), if P is a vertex or the midpoint or a midpointof an edge of Kljk e C®,

f fzhds = z ds , if S is a side of X I j k G Cft° ,

Js Js

where wh e Vh satisfies

(Vz - Wwh, Vi?) = 0 Vi; G Ffc .

Using the same argument as in [5, pp. 76-77] one can verify that zh is welldefined and that

(divz„, nJ = (divz,m).

Thus we have

(H^divzJ Gi^divz) ^ ^ ., „

which proves the lemma. D

vol 16, n° 3, 1982

282 J. PITKÀRANTA

Remark : In the argument of [5] referred to above one assumes that theLaplacian is an isomorphism from H2(Q) n H J(Q) to L2(Q). This obviouslyholds in the present case. D

Proofof Lemma 3.1 : Let |i = £ aljkl ÇyW = £ ji, be given with (n, 1) = 0.ijkl l

We first define the functions z = (z1, z2 , z3) eVh,w = (w l5 w2 , w3) e F h and

0 = (0i , 02, 03) e fc a s follows :

r Zi(P) = - hocljk2

(i) J z2(P) = — haljk3 if P is the midpoint

[z3(P)= - K M oîKljkeC°

(ii) w3(P) = - h{aljk5 - ccld+Uk5l or respectively

w2(P) = - /z(aljk5 - a l J t k + 1 ,5) ,

if P is the midpoint of the common side oîKljk and KltJ+ ltk e Ch°,or of Kljk and KlJtk+1eC£,

(iii) w 3 ( P ) = - fi(a I j k6 - OL1+1IJ6) , or respect ively

W l ( P ) = - h(aljk6 - a l J f f c + 1 > 6 ) ,

if P is t h e m i d p o i n t of t h e c o m m o n s ide of Kljk a n d Kl+ ljk G Q ° ,or oïKljk

(iv) w2(P) = - h(ocljkl - Gcl+Ujk7), or respectively

Wi(P) = - fc(ayk7 - a l t J+l fJk7),

if P is the midpoint of the common side oîKljk and X l + 1>jk e

°

= M " XijkS + ai+l,jlk8 + a^+l,*8 - «i+Lj+lJksh O r

= / i ( - aljfc8 + aI+1>jfc8 + a l M + l 5 8 - a I + 1 > j 5 k + l j 8 ) , or

if, respectively, P is the midpoint of the common edge of

and Kl+lfJ+uke C£, or of

Ch9 or of

(vi) The remaining degrees of freedom of z, w and # are set equal to zero.

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THE STOKES PROBLEM IN IR3 283

One can easily verify from (i) through (vi) that the following inequalitieshold :

1/2

1/2

gh ^ Ch3'2

jlo h

8 \ / 7£ \il9 div w ^ ChH £

1=5 / \I=5

and

(\ix + p8 , divg) ^ Ch3 a(p8)2 .

We now introducé a fourth function e = (eu e2, e3) e Vh which satisfu-s

(|i1?div^) ^ C || M-i Ho-

Since (p, 1) = (p l5 1) = 0, the existence of e follows from Lemma 3.2.Now, let v = z + 8w + b2g + 53e, where 5 e [0, 1] will be chosen below.

Then we have

IMIi < C | * i | f c , (3.1)

and

(p, div t;) > C ^ 5 3 || px ||g + £ II M-i II o + 8h )L a ( ^ ) + " ^

+ 8 X (M'Ï' ^ V W )1=2

7

+ 52 E (to'àivg)1 = 2

8

+ 5 3 Y (Vb div e) • (3 • 2)J = 2

We will now deed estimâtes for | (pz, div gf) | and | (p , div e) \ for / = 5,..., 8.We proceed as follows. For v = (vl9 v2, v3) e Vh9 let

v„ijk = vn{ihi \% jh2/2, kh3/2),

vol. 16, n° 3, 1982

284 J. PITKÂRANTA

i = O,..., 2 ml9j = O,..., 2 m2, k = O,..., 2 m3, n = 1, 2, 3. Then we can write(|i,, div v\ l = 5,..., 8, v e F,,, explicitly in terms of aljkl and t;nijfc. For exam-ple, we find by straightforward computation that

mi— 1

(]Ll5, div V) = £ E K*5 - <**,;+ 1,*5) A7 =1 i,k

where

16 v=o

and

16 x 3 ^1 0 i = o

2

Z Cl(V2i-lt2j-2,2k ~

— T7ni n2 ZJ CAV2i-l,2j,2k-2 ~ ^V2i-l,2j,2k~l + V2i-l,2j,2k) ->1 0 J = 0

where c0 = 1, c t = 2 and c2 = 1.Similarly, we find that

s^-p—J=j^h2h3lJ 2- Z Ay^J xCAX^y 1D ! J = 1 fc=1

X (aijfc8 ~ ai,j+l,k8 "~ aij,fc+l,8 + ai,j+l,Jk+l,8) ?

where

Ajjfcfa) = ^2i-2,2j2k "~ 2^2i-l,2j,2k + Ü2i,2j,2k •

Using these relations and similar expressions for (|i6, div v\ ((i7, div v\(|a8, dv2/dx2) and (|i8, dv3/dx3\ and noting that

3 2 m i - l 2 m 2 - l 2 m 3 - l

C i \2 u X"1 \~* \^ X"1 r \21 | ü 11 ^ « 2^ 2^ 2^ 2J U nyfc ~~ ün,i+l,jk) +

^ C 2 | V \\ , ü 6 Ffc ,

we can now easily verify that

(vLl)\v\i, / = 5 ,6 ,7 , p e n , (3.3)

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THE STOKES PROBLEM IN R 3 285

and

8 , div v) | ^ Ch3/2 a(ji8) | v | i , veVh. (3.4)

Applying (3.3) and (3.4) together with the above estimâtes for || w || l5g || ! and || e || x in (3.2) we find that

ftdivt;)^c{ô3||^||2+ t IIMo + ö/z3 £ a(ft)2+52/i3a(^)2j -l 1=2 1=5 J

4 ") 1/2 r 7 ") 1/2

4- 7

r ii ft ii o + h3 YJ C

r 4 8 ^)l/2- C , 5 3 I B M , IIS+ ^ 1 0 0 1 , ) -

(C - C 2 Ô) { Ô3 II Ui ||g + £ II H, ||g + 5/t3 Xi = 2 ï = 5

S 2 / i 3 aOi 8 )

. f C lChoosing now 5 = min < 1, >, we have

l 2Ü J(ji, div v) ^ C | ^ |2 .

Together with (3.1), this proves the asserted lower bound for | ju \h. To finallyprove the upper bound we only need to note that, by (3.3) and (3.4),

| (ft div v) | ^ C | n IJ v |i , [i e Qh, i? e 7fc .

Thus, Lemma 3.1 is proved. DWe note that, by the définition of Nf, | . \h is a norm in JVfc

x. We establishnext a lower bound for this norm in terms of h and the usual Lp norms.

LEMMA 3.3 : If \i s Njj-, then

\i k > C ( t II Vi IIO + * Z II Mi IIO + ^5/2 II 8 110,61 .

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286 J. PITKÀRANTA

Proof : Let \i — £ aljkl L,ljkl e N^ be given. We recall from the définition

of Nh that £ aijk,5 — Z aijk,6 = YJ aijfe,7 = 0. From these relations wej k i,k ij

conclude, e g , that

Uk

J * = 1

.2 \ r< u2 H .. n 2ijk5)(a l j k 5)2 ^ Cx /z2 || u.5 \\l

Here we used discrete Poincare's and Sobolev's mequahties to conclude thatif Z otjk = 0, then

Jk

i— 1

l Ek

(c/ [7] for the details of the argument) Since similar estimâtes obviously holdfor a(|i6) and a(u7), we conclude that

t \\Vi\\o + h I I I ^ H o V (3 5)Z = 1 / = 5 /

To obtain a bound for the component u8 = £ aIjk8 ^uk8, let k be fixed,

< fe < m3 — 1, and defïne

Then we easily find that

i - i

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THE STOKES PROBLEM IN R 3 287

Recallmg that Z aijk8 = 0 for fe = 1,. ., m3 (since \i e N^\ we have m parti-

cular that Z StJ = 0 Usmg this we may solve for 8 l s l m (3 6) to obtain

m i — 1 nt2— 1

1=1 l j = l

where the coefficients satisfy

\Cj\<C

\dtJ\ ^ Ch.

Substituting this back to (3 6) we obtain

(m i - l m 2 - l \

E EP5 + I I Y5). o?)i=i j » J = I /

Repeating this argument for all k and for permuted indices, and summmgup the resultmg inequalities (3 7), we find that

a ü i 8 ) > C h | m l i * , (3 8)

where

m i — 1

ZJ Z (aijfc8 "" ai+l,jfc8) +

nt2 1 WI3 1\2

2 a

E ZKfcs - <*i,j+i,ks)2 + Zj=i a k=i

- aIJfJk+1>8)

T o finally get a lower b o u n d for | \i8 \lth we construct a fonction cp e H ^satisfying

oo

and

cp dx = hx h2 h3 Z ccljkS = 0 .Jn ï.j.k

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288 J. PITKÂRANTA

The fonction cp is found, e.g. as follows. Consider another rectangular sub-division Ch of Q, the interior nodes of which are located at the midpoints ofKijk G Ch- Then defïne cp to be the continuous piecewise trilinear function onQ,1, which satisfies cp(x) = aijk8 if x is a node of C* such that x e Kijk, Kijk e Ch°.It is then easy to see that the above relations hold, and so, using Poincare'sand Sobolev's inequalities, we find that

IHsli.* > C|<Pli > Cx || cpili > C2 | |q>||o f6

> C3 || ^8 II 0,6 •

Combining this with (3.8) and recalling the définition of | |! \h we obtain

Together with (3.5) this finishes the proof of Lemma 3.3. DWe can now state and prove a basic error estimate for the scheme (2.2).

THEOREM 3 . 1 : Assume that the solution of(2A) satisfies

(u,X)e[W9/2>6l5(Q)f x H^Q).

Then if(uh Xh) eVhx N^~ is a solution to (2.2) and X is the orthogonal projec-tion of X onto AT,,1, we have

u - uh\x + | Xh - Xh\h^ Ch{\\ u9/2f6/5

Proof : Let ü e Vh be the interpolant of u. We first apply Lemma 3.1 andthe gênerai theory of Babuska [1] and Brezzi [2] (cf. also [7]) to conclude theexistence of (v, ji) e Vh x N^ such that

I » li + I \i \h < C,and

\uh-ü\1 + \Xh-X\h^C{\ (V(ii - ü\ Vv) | +

+ | (X - X, div v) | + | (div(u - S), |i) I } . (3.9)

The first term on the right side of (3.9) obeys as usual (cf. [3]) the quasi-optimal bound

| (II - ü\ Vv) | ^ | u - ü \x | v \x ^ Ch | u \2 . (3.10)

The second term can be estimated by first noting that

(X, div v) = (nh X, div v) VveVh9

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THE STOKES PROBLEM IN 289

where nh X is the orthogonal projection onto Qh. Hence, by well-knownapproximation theory,

- « ; div t;) (3.11)

In estimating the third term on the right side of (3.9) we need the following« superapproximation » result, the proof of which is straightforward.

LEMMA3.4 : Defining for v e [H2(K)f,K = KlJkeCh°,

JK lJkl

where v dénotes the piecewise trilinear interpolant oj v on the eight subrectanglesof K, we have

and

L, (v) = 0 , / = 1,..., 8 , if ve [P2]3

L8(i>) = 0, if V€[P5]3,

so that, in particular,

and

Ls(v) | < Chk+2~3/p | v \wuP{K),

Now writing |i = ]T <xljkl ^ljkl = £ \ix we have

4

(div (u — ü\ Y, Hz) ^ C | w — w |x

< C± fc | u |2 ,

and, applying Lemma 3.4 and Lemma 3.3,

(3.12)

(div (u —

vol 16, n°3,

7

ü\ I Hz)

1982

^Ch2\

hJ,k

U 3

7 Ç

1=5 JK

i

1=5

div (u — ü) ^ljki dx

« C , » i . u . (3.13)

290 J PITKARANTA

Similarly, applying the Holder mequahty and Lemma 3 4 we find that

| ( d i v ( W - u ) , M | ^ C t f - M a l k p l I M o , ,

l ^ p < o o , p-1 + q~x = 1, 4 ^ k ^ 6 (314)

Choosing here p = 6/5, we have q = 6 and so, by Lemma 3 3,

llmllo,<cfc ^ I H I ^ Q / T 5 ' 2

By mterpolatmg in (3 14) we then obtain

\(div(u-ü\ii8)\ < C f c 7 ' 2 H u | | 9 / 2 6 / 5 | | j i | | 0 6

^ C1 h II u H 9/2 e/s (3 15)

From (3 12), (3 13) and (3 15) we see, applying the Sobolev embeddmg,that

|(div(w - M), \I)\ ^ Ch\\u | |9 /26 /5

Combming this with (3 9) through (3 11) and finally applying the trianglemequahty together with the usual bound for | u — ü |l5 we obtain the desiredestimâtes for | u — uh \1 and | Xh — X \h and the proof of Theorem 3 1 iscomplete D

Remark The regulanty assumption in Theorem 3 1 is not quite reahsticeven in the simple geometry considered, since there are in gênerai singularitiesm the solution near the adges and vertices of Q Taking the leading edge sin-gulalanty into account, we conjecture from [6, 9] that u can satisfy

ue[Ws65(Q)f for s < 4,4

if ƒ in (2 1) is sufficiently smooth With this regulanty assumption, we wouldobtain || ii - HJII « 0(/z09) D

Remark One cannot obtain any convergence rate for the pressure in L2

from Theorem 3 1, since Lemma 3 3 only implies that

However, as in [7], ît follows easily from the définition of | • \h that if Xh is firstaveraged over each Kljk e Ch° then the resultmg smoothed pressure 7t£ Xh

converges

|l9/2 6/5 + ll^lll) •

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THE STOKES PROBLEM IN R 3 291

Remark : Assuming that we have for Eq. (2.1) the a priori estimate

\ \ u \ \ i + II M i < C \ \ f H o ,

which is generally conjecturée for a convex polyhedral domain, one can proveusing the technique of [7] that

| | u - u J 0 < C f c 2 ( | | t i | | 9 / 2 f 6 / 5 + \\X\\t). D

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